For (REF ) one can rely on Hölder's inequality, while (REF ) follows from the inclusion \(\ell ^{q_{1}}\subseteq \ell ^{q_{2}}\) . Moreover, both (REF ) and (REF ) follow from standard embeddings between Besov and Sobolev spaces (see [1]}).
Finally, by Proposition REF , for (REF ) it suffices to show that
\(\Vert \chi _{\nu }^{k}(D)f\Vert _{L^{p_{2}}(\mathbb {R}^{d})}\lesssim 2^{k\frac{d+1}{2}(\frac{1}{p_{1}}-\frac{1}{p_{2}})}\Vert \chi _{k}(D)f\Vert _{L^{p_{1}}(\mathbb {R}^{d})}\) 
for all \(k\ge 0\) , \(\nu \in \Theta _{k}\)  and \(f\in \mathcal {B}^{s}_{p_{1},q,q}(\mathbb {R}^{d})\) . To this end, one can construct a collection \(\lbrace \tilde{\chi }_{\nu }^{k}\mid k\ge 0,\nu \in \Theta _{k}\rbrace \subseteq C^{\infty }_{c}(\mathbb {R}^{d})\)  of cut-offs with similar support and decay properties as the \(\chi _{\nu }^{k}\) , but such that \(\tilde{\chi }_{\nu }^{k}\equiv 1\)  on \(\mathrm {supp}(\chi _{\nu }^{k})\) . Then, as in (REF ),
\(\Vert {\mathcal {F}}^{-1}(\tilde{\chi }_{\nu }^{k})\Vert _{L^{p_{3}}(\mathbb {R}^{d})}\lesssim 2^{k\frac{d+1}{2}(\frac{1}{p_{1}}-\frac{1}{p_{2}})},\) 
where \(\frac{1}{p_{3}}=1+\frac{1}{p_{2}}-\frac{1}{p_{1}}\) . Now (REF ) follows from Young's inequality, since \(\chi _{\nu }^{k}=\tilde{\chi }_{\nu }^{k}\chi _{\nu }^{k}\) .
